Question: Simplify and expand the following expression: $ \dfrac{5}{k - 9}- \dfrac{4}{k + 2}- \dfrac{5k}{k^2 - 7k - 18} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{5k}{k^2 - 7k - 18} = \dfrac{5k}{(k - 9)(k + 2)}$ Now we have: $ \dfrac{5}{k - 9}- \dfrac{4}{k + 2}- \dfrac{5k}{(k - 9)(k + 2)} $ The least common multiple of the denominators is: $ (k - 9)(k + 2)$ In order to get the first term over $(k - 9)(k + 2)$ , multiply by $\dfrac{k + 2}{k + 2}$ $ \dfrac{5}{k - 9} \times \dfrac{k + 2}{k + 2} = \dfrac{5(k + 2)}{(k - 9)(k + 2)} $ In order to get the second term over $(k - 9)(k + 2)$ , multiply by $\dfrac{k - 9}{k - 9}$ $ \dfrac{4}{k + 2} \times \dfrac{k - 9}{k - 9} = \dfrac{4(k - 9)}{(k - 9)(k + 2)} $ Now we have: $ \dfrac{5(k + 2)}{(k - 9)(k + 2)} - \dfrac{4(k - 9)}{(k - 9)(k + 2)} - \dfrac{5k}{(k - 9)(k + 2)} $ $ = \dfrac{ 5(k + 2) - 4(k - 9) - 5k} {(k - 9)(k + 2)} $ Expand: $ = \dfrac{5k + 10 - 4k + 36 - 5k}{k^2 - 7k - 18} $ $ = \dfrac{-4k + 46}{k^2 - 7k - 18}$